Abstract

The paper contains some suffcient conditions for the existence of positive solutions which are bounded below and above by positive functions for the nonlinear neutral differential equations of higher order. These equations can also support the existence of positive solutions approaching zero at infinity.

1. Introduction

This paper is concerned with the existence of a positive solution of the neutral differential equations of the form:
where is an integer, , , , , , is a nondecreasing function and , .

By a solution of (1.1) we mean a function for some , such that is -times continuously differentiable on and such that (1.1) is satisfied for .

The problem of the existence of solutions of neutral differential equations has been studied and discussed by several authors in the recent years. For related results we refer the reader to [1–17] and the references cited therein. However, there is no conception which guarantees the existence of positive solutions which are bounded below and above by positive functions. Maybe it is due to the technical difficulties arising in the analysis of the problem. In this paper we presented some conception. The method also supports the existence of positive solutions which approaching zero at infinity. Some examples illustrating the results.

The existence and asymptotic behavior of solutions of the nonlinear neutral differential equations and systems have been also solved in [1–7, 12, 15].

As much as we know for (1.1) in the literature, there is no result for the existence of solutions which are bounded by positive functions. Only the existence of solutions which are bounded by constants is treated and discussed, for example, in [10, 15, 17]. It seems that conditions of theorems are rather complicate, but cannot be simpler due to Corollaries 2.4, 2.8, and 3.3.

The following fixed point theorem will be used to prove the main results in the next section.

Lemma 1.1 (see [7, 10, 12] Krasnoselskii’s fixed point theorem). Let be a Banach space, let be a bounded closed convex subset of , and let be maps of into such that for every pair . If is a contractive and is completely continuous then the equation:
has a solution in .

2. The Existence of Positive Solution

In this section, we will consider the existence of a positive solution for (1.1) which is bounded by two positive functions. We will use the notation .

Theorem 2.1. Suppose that there exist bounded functions , constant , and such that
Then (1.1) has a positive solution which is bounded by the functions , .

Proof. Let be the set of all continuous bounded functions with the norm . Then is a Banach space. We define a closed, bounded, and convex subset of as follows:
We now define two maps and as follows:
We will show that for any we have . For every and we obtain
For we have
Furthermore for we get
Finally let and with regard to (2.2) we get
Then for and any we get
Thus, we have proved that for any . We will show that is a contraction mapping on . For and we have
This implies that
Also for the inequality above is valid. We conclude that is a contraction mapping on . We now show that is completely continuous. First we will show that is continuous. Let be such that as . Because is closed, . For we have
According to (2.8) we get
Since as , by applying the Lebesgue dominated convergence theorem we obtain that
This means that is continuous. We now show that is relatively compact. It is sufficient to show by the Arzela-Ascoli theorem that the family of functions is uniformly bounded and equicontinuous on . The uniform boundedness follows from the definition of . For the equicontinuity we only need to show, according to Levitan result [8], that for any given the interval can be decomposed into finite subintervals in such a way that on each subinterval all functions of the family have change of amplitude less than . With regard to the condition (2.14), for and any we take large enough so that
Then for we have
For , and we get
With regard to the condition (2.14) we have that
Then we obtain
Thus there exists a , where
such that
For we proceed by the similar way as above. Finally for any , there exists a such that
Then is uniformly bounded and equicontinuous on and hence is relatively compact subset of . By Lemma 1.1 there is an such that . We conclude that is a positive solution of (1.1). The proof is complete.

Corollary 2.2. Suppose that all conditions of Theorem 2.1 are satisfied and
Then (1.1) has a positive solution which tends to zero.

Corollary 2.3. Suppose that there exist bounded functions , constant and such that (2.1), (2.3) hold and
Then (1.1) has a positive solution which is bounded by the functions .

Proof. We only need to prove that condition (2.25) implies (2.2). Let and set
Then with regard to (2.25), it follows that , . Since and for , this implies that
Thus all conditions of Theorem 2.1 are satisfied.

Corollary 2.4. Suppose that there exists a bounded function , constant and such that
Then (1.1) has a solution .

Theorem 2.5. Suppose that is bounded and there exist bounded functions , constant and such that (2.1), (2.2) hold and
if is odd,
if is even, and
Then (1.1) has a positive solution which is bounded by the functions , .

Proof. Let be the set as in the proof of Theorem 2.1. We define a closed, bounded, and convex subset of as in the proof of Theorem 2.1. We define two maps and as follows:
We shall show that for any we have . For odd, every and we obtain
For , we have
Furthermore for , we get
Let and according to (2.2) we have
Then for and any we get
Thus we have proved that for any . For even by the similar way as above we can prove that for any .As in the proof of Theorem 2.1, we can show that is a contraction mapping on . We now show that is completely continuous. First, we will show that is continuous. Let be such that as . Because is closed, . For we have
According to (2.33) there exists a positive constant such that
The inequality above also holds for even.Since as , by applying the Lebesgue dominated convergence theorem we obtain that
This means that is continuous. We now show that is relatively compact. It is sufficient to show by the Arzela-Ascoli theorem that the family of functions is uniformly bounded and equicontinuous on . The uniform boundedness follows from the definition of . For and with regard to (2.31) we have
and for we obtain
for , and for , , which shows the equicontinuity of the family , (cf. [7, page 265]). Hence is relatively compact and therefore is completely continuous. By Lemma 1.1, there is such that . Thus is a positive solution of (1.1). The proof is complete.

Corollary 2.6. Suppose that all conditions of Theorem 2.5 are satisfied and
Then (1.1) has a positive solution which tends to zero.

Corollary 2.7. Suppose that is bounded and there exist bounded functions , constant and such that (2.1), (2.29), (2.30), (2.31) hold and
Then (1.1) has a positive solution which is bounded by the functions .

Example 3.4. Consider the nonlinear neutral differential equation:
where . We will show that the conditions of Theorem 3.1 are satisfied. The condition (3.1) has a form:
,. For function , we obtain
For , the condition (3.9) is satisfied and
If the function satisfies (3.11), then (3.8) has a solution which is bounded by the functions .

Example 3.5. Consider the nonlinear differential equation:
where , and
for . Set
Then we have
By Corollary 2.7, (3.12) has a solution which is bounded by the functions and . If
then (3.12) has the positive periodic solution .

Acknowledgment

The research was supported by the Grants 1/0090/09 and 1/1260/12 of the Scientific Grant Agency of the Ministry of Education of the Slovak Republic.